Internat. J. Math. & Math. Sci.
VOL. 18 NO. 2 (1995) 411-415
411
COMPUTATIONS FOR A VIBRATING SYSTEM DIAGONALIZE THE VARIANCE
J. N. BOYD and R N. RAYCHOWDHURY
Department of Mathematical Sciences
Virginia Commonwealth University
Richmond Virginia 23284-2014
(Received August II, 1993 and in revised form September 15, 1993)
ABSTRACT. The transformations to diagonalize potential energy matrices for coupled harmonic
oscillators will also diagonalize the variance when written in matrix form. After a brief review of
a geometrical interpretation of the variance, the transformations are described and an example is
given.
KEY WORDS AND PHRASES. Variance, orthogonal and unitary transformations, coupled
oscillators, elastic potential energy.
1991 AMS Subject Classification Code. Primary 20C99, Secondary 20C35.
INTRODUCTION. In this note, we will be interested in n independently chosen real numbers
x,,...,x
with mean x
x
and in their variance
which can also be written as
()
We begin by recalling a geometric interpretation for s2.
Let the n-tuple (x,x2, ...,xn) represent a point P in n-dimensional Euclidean space (E(n))
al,a2 a
and let orthogonal unit vectors
be defined in the positive
e=xa
+xna
respectively. Then the vector
+x2a2 +...
from the origin O:(0,0 0) and to terminate at P.
directions,
can be taken to emanate
Th.e projection of e upon the line t:x, =x =... =x
’(al+a2+’’’+an)’
qr
nn-11x
x,xv...,x,,
is given by the inner product
as suggested in Figure 1. In n-dimensions, the distance
(d) from P to t satisfies the equation
i-I
(2)
n
Thus s is simply proportional to the square of the distance from P to
the simplification, we first rewrite Equation 1 in matrix form:
e. To pursue
J. N. BOYD AND P. N. RAYCHOWDHURY
412
Figure 1: The Geometry of P, 7, and e.
Our problem is to diagonalize the matrix.
-
-
(3)
AN EXAMPLE. To illustrate the calculations, let us consider the case for n
x,,x2,x,x 3s
(x xxx4)
3 -1 -1 -1
3 -1 -1
-1
3
-I -I
xl
x2
x
x
4
4. Then
l
I
In E(4), we imagine surfaces of constant :(xl, x2,x,x4) to be concentric cylinders with
It is clear that in choosing new coordinates, we must take one coordinate to be
+ x2 +
+ x4 to place one axis on e. Simple choices of the other three
proportional to
are proportional to
v x-x4 and -x-x2 +x3 +x to determine the orthogonal
x
x
x-x
transformation matrix
0
0
0
2
2
1
2
2
2
2
0
2
1
2
(4)
COMPUTATIONS FOR A VIBRATING SYSTEM DIAGONALIZE THE VARIANCE
413
We can rewrite Equation 4 as
4
(xt x2xsx4)$ -IS
3 -1 -1 -1
-1 3 -1
$
-1 3
-1 -1
xt
x2
x
00
10
+
where the new coordinates
x, x2, x
+
/2
combine to yield the square of the distance from
which
is now the x4 axis. The appearance of only three summands is consistent with the
4 1 3 degrees of freedom for the variance.
ANOTHER APPROACH. Let us now return to Equation 1. After a bit of algebra, we
see that the equation for the variance can be rewritten yet again as
s2
1_
n-1
1
n
.
.m
(x_x)2.
Having obtained Equation 5, we recognized it to be essentially the same as that for the
elastic potential energy for n masses symmetrically arranged on a f’Lxed circle and
interconnected by idealized harmonic springs between all pairs of masses. Furthermore, we
have previously described this system.[ 1]
Returning to the case n
in Figure 2.
4, let us consider the four masses on the circle as shown
Xl
Figure 2. The Coupled Oscillators.
The particles are constrained to move on the circle and their displacements from
(5)
414
-
J. N. BOYD AND P. N. RAYCHOWDHURY
their equilibrium positions are taken to be Xl, x2, x3, x4. We choose the elastic constants for
all couplings to be k. Then the potential energy can be written as
P.E.
:[<x
-, )2
)2 +
/
x
x )2 +
x,)
.) +
/
, )].
The symmetries which simplified the potential energy function were derived from the
rotations of the circle. The same symmetries can be applied to f(xl, x2, x3, x4) of
Equation 4. The four-by-four matrix
3 -1 -1 -1
3
-1
-1 -1
can be diagonalized by the unitary transformation having matrix representation
-1 -i
1
-i -1
which was computed from the irreducible representations of the group of rotations by 90*
180 270 and 360* about the center of the circle of Figure 2. [2,3]
, ,
Although the transformation carries the coordinates into complex numbers, f which
is proportional to the variance remains fixed and real:
f(Xl’X2’X"X4)
=-
(x
x4)U
I-
3 -1
3 -I -1
U_U x2
-1
-1
11
1 -1
3
x4
40
where Y,, Yi, and r
represent the new normal coordinates, their complex conjugates, and
Thus the variance has again been diagonalized. In addition, we
now have unitary matrices to diagonalize the variance matrix for all n > 2. They have
their moduli, respectively.
already been computed to diagonalize potential energies.
GENERALIZATION. The purpose of this note is to point out that the unitary transformations
developed from the group of rotations of the circle enable us to look at variance and
COMPUTATIONS FOR A VIBRATING SYSTEM DIAGONALIZE THE VARIANCE
415
standard deviation. We had previously used these transformations to find natural
frequencies for oscillating mass and spring systems. We were pleased that previous work
solved a new problem for us. We simply state the generalization. The variance for n _> 2 real
numbers x,x2,...,x,, can be computed as
s2
(x)2
n-1
1
!i
X
-1 -1
n-1 -1
-1
-1
x
1
1
x2.
(6)
n-1
The unitary matrix
U= 1
2hi
4hi
exp 6hi
exp
exp--
exp 2(n-1)ni 1
12i
exp 4(n-1)ni 1
8i
4hi
exp exp-- exp
1
1
1
1
1
which was developed from the representations of the Abelian group of rotations through
2--En
of
the circle in order to diagonalize a potential energy matrix will also diagonalize the variance matrix.
REFERENCES
BOYD, J. N. and RAYCHOWDHURY, P.N., A Group Theoretic Approach to Generalized
Harmonic Vibrations in a One Dimensional Lattice, Int. J, of Math, nd Math. Sci. 9 (1986),
131-136.
BOYD, J. N. and RAYCHOWDHURY, P.N., An Application of Projection Operators to
a One Dimensional Crystal, Bull, of the Inst, of Moth,, Academia $inica 7 (1979), 133-144.
BOYD, J. N. and RAYCHOWDHURY, P.N., Group Representations in Lagrangian
Mechanics, Physica l14A (1982), 604-608.